An important motivation for (REF ) arises from pure states in Lieb-Thirring estimates and interpolation inequalities for systems studied in [1]}. We refer to [2]} for an overview of functional inequalities and branches of solutions in various frameworks, including the quite similar problem of symmetry breaking in some of the (CKN) Caffarelli-Kohn-Nirenberg inequalities. The sharp threshold for the symmetry range of the parameters in such inequalities is established in [3]}, [4]} in the critical case and in [5]}, [6]} in the subcritical case. A complete parabolic proof based on entropy methods is so far missing and results rely on rigidity methods: see [7]}, [8]} for partial results based on flows. In the case of the sphere, Inequalities (REF ), (REF ) and (REF ) have similar properties but by many aspects symmetry versus symmetry breaking issues are simpler. For instance, the regularity of the solution of the nonlinear flow on \({\mathbb {S}}^d\)  raises no difficulty: see for instance [9]} for some comments in this direction. Branches of optimizers associated with Inequalities (REF ) and (REF ) for \(\theta <1\)  have qualitative features which are reminiscent of those corresponding to similar (CKN) inequalities (with also some \(\theta <1\) ) observed in [10]}, [11]}, [12]}. Beyond numerical observations and a few estimates, these branches for (CKN) are not well understood. Our goal is to state the corresponding properties on the sphere.
